Generally, for every Lie-∞-algebroid𝔞\mathfrak{a} one may define the corresponding tangent Lie-∞\infty-algebroid T𝔞T \mathfrak{a}, whose Chevalley-Eilenberg algebra may be called the Weil algebra of 𝔞\mathfrak{a}:

W(𝔞)=CE(T𝔞).
W(\mathfrak{a})
=
CE(T \mathfrak{a})
\,.

Weil algebra of a Lie algebra

Let 𝔤\mathfrak{g} be a finite-dimensional Lie algebra. The Weil algebraW(𝔤)W(\mathfrak{g}) of 𝔤\mathfrak{g} is

the graded Grassmann algebra generated from the dual vector space𝔤*\mathfrak{g}^* together with another copy of 𝔤*\mathfrak{g}^* shifted in degree

equipped with a derivationd:W(𝔤)→W(𝔤)d : W(\mathfrak{g}) \to W(\mathfrak{g}) that makes this a dg-algebra, defined by the fact that on 𝔤*\mathfrak{g}^* it acts as the differential of the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g} plus the degree shift morphism 𝔤*→𝔤*\mathfrak{g}^* \to \mathfrak{g}^*.

This Weil algebra has trivial cohomology everywhere (except in degree 0 of course) and sits in a sequence

Definition

For L∞L_\infty-algebras

Let 𝔤\mathfrak{g} be an L-∞ algebra of finite type. By our grading conventions this means that the graded vector space𝔤*\mathfrak{g}^* obtained by degreewise dualization is in non-negative degree, and ∧1𝔤*=𝔤*[1]\wedge^1 \mathfrak{g}^* = \mathfrak{g}^*[1] is its shift up into positive degree.

Definition

The Weil algebraW(𝔤)W(\mathfrak{g}) is the unique representative of the freedg-algebra on ∧1𝔤*\wedge^1 \mathfrak{g}^* for which the projection of graded vector spaces ∧1(𝔤*⊕𝔤*[1])→∧1𝔤*\wedge^1(\mathfrak{g}^* \oplus \mathfrak{g}^*[1]) \to \wedge^1 \mathfrak{g}^* extended to a dg-algebrahomomorphismW(𝔤)→CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g})

We discuss below in the Properties section that this is equivalent to the following component-wise definition

d\mathbf{d} acts by degree shift 𝔤*→𝔤*[1]\mathfrak{g}^* \to \mathfrak{g}^*[1] on elements in 𝔤*\mathfrak{g}^* and by 0 on elements of 𝔤*[1]\mathfrak{g}^*[1];

dCE(𝔤)d_{CE(\mathfrak{g})} acts on unshifted elements in 𝔤*\mathfrak{g}^* as the differential of the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g} and is extended uniquely to shifted generators by graded-commutativity

Definition

The Weil algebraW(𝔞)W(\mathfrak{a}) of the L∞L_\infty-algebroid 𝔞\mathfrak{a} is the Chevalley-Eilenberg algebra of the L∞L_\infty-algebroid defined as follows

the TT-algebra AA in degree 0 is the same as that of 𝔄\mathfrak{A};

the underlying graded algebra is the exterior algebra on 𝔞*\mathfrak{a}^* and a shifted copy 𝔞*[1]\mathfrak{a}^*[1] as well as one copy of the Kähler differential module Ω1\Omega^1 in lowest degree (though of as the shifted copy of AA itself)

of two degree +1 graded derivations, where dCE(𝔞)d_{CE(\mathfrak{a})} and a\mathbf{a} are defined on ∧1𝔞*⊕𝔞*[1]\wedge^1 \mathfrak{a}^* \oplus \mathfrak{a}^*[1] as above for L∞L_\infty-algebras and on AA itself dCE(𝔞)d_{CE(\mathfrak{a})} vanishes and d\mathbf{d} acts as the universal derivation

By solving the condition dW(𝔞)∘dW(𝔞)=0d_{W(\mathfrak{a})} \circ d_{W(\mathfrak{a})} = 0 and using that dCE(𝔞)dCE(𝔞)=0d_{CE(\mathfrak{a})} d_{CE(\mathfrak{a})} = 0 this already fixes uniquely the differential dW(𝔞)d_{W(\mathfrak{a})}. To see this we only need to show that the value of dW(𝔞)(x)d_{W(\mathfrak{a})}(x) on a generator x=σ(t)∈𝔞*[1]x=\sigma(t) \in \mathfrak{a}^*[1] is completely determined by dW(𝔞)|∧•𝔞*d_{W(\mathfrak{a})}\vert_{\wedge^\bullet\mathfrak{a}^*}. One computes:

To prove injectivity, we just have to show that the restriction of a dgca morphism f:W(𝔤)→Af:W(\mathfrak{g})\to A to 𝔤*\mathfrak{g}^* determines the restriction of ff to 𝔤*[1]\mathfrak{g}^*[1]. One has, for any x=σ(t)∈𝔤*[1]x=\sigma(t)\in \mathfrak{g}^*[1],

Since dCE(𝔤)(t)d_{CE(\mathfrak{g})}(t) lies in the sub-gca of W(𝔤)W(\mathfrak{g}) generated by 𝔤*\mathfrak{g}^*, the element f(dCE(𝔤)(t))f(d_{CE(\mathfrak{g})}(t)), and therefore f(x)f(x), is determined by f|𝔤*f\vert_{\mathfrak{g}^*}.

Next we show surjectivity, i.e. that every morphism of graded vector spaces ϕ:𝔤*→A\phi:\mathfrak{g}^*\to A can be extended to a dgca morphism f:W(𝔤)→Af:W(\mathfrak{g})\to A. Denote by f0:∧•𝔤*→Af_0: \wedge^\bullet \mathfrak{g}^*\to A the extension of ϕ\phi to a graded commutative algebra morphism, and let ψ:𝔤*[1]→A\psi:\mathfrak{g}^*[1]\to A be the graded vector space morphism defined by

for any x=σ(t)∈𝔤*[1]x=\sigma(t)\in \mathfrak{g}^*[1]. The graded vector space morphism ϕ+ψ:𝔤*⊕𝔤*[1]→A\phi+\psi:\mathfrak{g}^*\oplus\mathfrak{g}^*[1]\to A extends to a commutative graded algebra f:W(𝔤)→Af:W(\mathfrak{g})\to A, whose restriction to 𝔤*\mathfrak{g}^* is ϕ\phi. We want to show that ff is actually a dgca morphism. We only need to test commutativity with the differentials on generators t∈𝔤*t\in \mathfrak{g}^* and x=σ(t)∈𝔤*[1]x=\sigma(t)\in \mathfrak{g}^*[1]. We have

Example

For A=CE(𝔤)A=CE(\mathfrak{g}) the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g}, the inclusion 𝔤*↪CE(𝔤)\mathfrak{g}^*\hookrightarrow CE(\mathfrak{g}) induces a canonical surjective dgca morphism W(𝔤)→CE(𝔤)W(\mathfrak{g})\to CE(\mathfrak{g}). This is the identity on the unshifted generators, and 0 on the shifted generators.

is the collection of total degree 1 differential forms with values in the ∞\infty-Lie algebra 𝔤\mathfrak{g}.

A morphism of

(A,FA):W(𝔤)→Ω•(X)
(A, F_A) : W(\mathfrak{g}) \to \Omega^\bullet(X)

sends the unshifted generators tat^a to differential forms AaA^a, which one thinks of as local connection forms, and sends the shifted generators σta\sigma t^a to their curvature. The respect for the differential on the shifted generators is the Bianchi identity on these curvatures.

A morphism W(𝔤)→Ω•(X)W(\mathfrak{g}) \to \Omega^\bullet(X) encodes a collection of flatL∞L_\infty-algebra valued forms precisely if it factors by the canonical morphism W(𝔤)→CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}) from above through the Chevalley-Eilenberg algebra of 𝔤\mathfrak{g}.

The freeness property of the Weil algebra can be made more explicit by exhibiting a concrete isomorphism to the free dg-algebra on 𝔤*\mathfrak{g}^*.

Definition

where the differential dfd_f is on the unshifted generators t∈𝔤*t \in \mathfrak{g}^* the shift isomorphism σ:𝔤*→𝔤*[1]\sigma : \mathfrak{g}^* \to \mathfrak{g}^*[1] extended as a derivation and vanishes on the shifted generators

dF:t↦σ(t),
d_F : t \mapsto \sigma(t)
\,,

dF:σ(t)↦0.
d_F : \sigma(t) \mapsto 0
\,.

Or in other words, if 𝔤¯\bar \mathfrak{g} is the ∞\infty-Lie algebra whose underlying graded vector space is that of 𝔤\mathfrak{g}, but all whose brackets vanish, then

Note that the isomorphism ff is precisely the dgca isomorphism induced between W(𝔤¯)W(\overline\mathfrak{g}) and W(𝔤)W(\mathfrak{g}) by the identity of 𝔤*\mathfrak{g}^* as a graded vector spaces morphism 𝔤¯*→𝔤*\overline{\mathfrak{g}}^*\to\mathfrak{g}^*.

Characterization in the smooth ∞\infty-topos

The Weil algebra of a Lie algebra is naturally identified with the de Rham algebra of differential forms on the “universal GG-principal bundle with connection” in its stacky incarnation (Freed-Hopkins 13):

The image of the unshifted generators A:∧1𝔤*→Ω•(X)A : \wedge^1 \mathfrak{g}^* \to \Omega^\bullet(X) are the forms themselves, the image of the shifted generators FA:∧1𝔤*[1]F_A : \wedge^1 \mathfrak{g}^*[1] are the corresponding curvatures. The respect for the differential on the shifted generators are the Bianchi identity on the curvatures.

Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebraW(𝔤)→CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}).

Invariant polynomials and Chern-Simons elements

An invariant polynomial⟨−⟩\langle -\rangle on 𝔤\mathfrak{g} is a closed element in the Weil algebra ⟨−⟩∈W(𝔤)\langle -\rangle \in W(\mathfrak{g}), subject to the additional condition that it its entirely in the shifted copy of 𝔤\mathfrak{g}, ⟨−⟩∈∧•(𝔤*[1])\langle - \rangle \in \wedge^\bullet (\mathfrak{g}^*[1]).

Examples

Weil algebra of a Lie algebra

Let 𝔤\mathfrak{g} be a finite dimensional Lie algebra. This Lie algebra regarded as a Lie algebroid has as base manifold the point, X0=ptX_0 = pt. Its algebra of functions is accordingly the ground field, and the algebra ∧C∞(X0)•𝔤*\wedge^\bullet_{C^\infty(X_0)} \mathfrak{g}^* is just a Grassmann algebra.

This also explains the use of the Weil algebra in the calculation of the equivariantde Rham cohomology of manifolds acted on by a compact group. These papers are reprinted, explained and put in a modern context in the book

A clasical textbook account of standard material is in chapter VI, vol III of

Some remarks on the notation there as compared to ours: our dWd_W is their δW\delta_W on p. 226 (vol III). Their δE\delta_E is our dCEd_{CE}. Their δθ\delta_\theta is our dρd_\rho (θ\theta/ρ\rho denoting the representation)..

The (obvious but conceptually important) observation that Lie algebra-valued 1-forms regarded as morphisms of graded vector spaces Ω•(X)←∧1𝔤*:A\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A are equivalently morphisms of dg-algebras out of the Weil algebra Ω•(X)←W(𝔤):A\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A and that one may think of as the identity W(𝔤)←W(𝔤):IdW(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id as the universal 𝔤\mathfrak{g}-connection appears in early articles for instance highlighted on p. 15 of